In the figure,$\angle AOB = 90^{\circ}$ and $\angle ABC = 30^{\circ},$ then $\angle CAO$ is equal to: (in $^{\circ}$)

In a circle with centre $P$,the diameter of the circle is $70 \, cm$. The chord $AB$ is at a distance of $21 \, cm$ from the centre of the circle,then $AB = \dots \, cm$.

The radius of a circle with centre $P$ is $50 \, cm$. Centre $P$ is not lying between two parallel chords $AB$ and $CD$. If $AB = 80 \, cm$ and the distance between $AB$ and $CD$ is $10 \, cm$,then find the length of $CD$. (Given $AB > CD$) (in $, cm$)

If two equal chords of a circle intersect,prove that the parts of one chord are separately equal to the parts of the other chord.

If $BM$ and $CN$ are the altitudes drawn on the sides $AC$ and $AB$ of the triangle $ABC$,prove that the points $B, C, M$ and $N$ are concyclic.

Two circles of radii $8 \ cm$ with centers $P$ and $Q$ are given. The chord $AB$ of the circle with center $P$ and the chord $CD$ of the circle with center $Q$ are equal. If $\angle PAB = 40^{\circ}$,then find $\angle CQD$. (in $^{\circ}$)